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Global solutions of a two-dimensional Riemann problem for the pressure gradient system

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    * Corresponding author 

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Gui-Qiang G. Chen's research was supported in part by the UK Engineering and Physical Sciences Research Council under Grant EP/L015811/1 and the Royal Society–Wolfson Research Merit Award WM090014 (UK). Qin Wang's research was supported in part by National Natural Science Foundation of China (11761077), China Scholarship Council (201807035046), and the Key Project of Yunnan Provincial Science and Technology Department and Yunnan University (No.2018FY001-014). Shengguo Zhu's research was supported in part by the Royal Society–Newton International Fellowships NF170015 and the Monash University–Robert Bartnik Visiting Fellowship. Qin Wang would also like to thank the hospitality and support of the Mathematical Institute, University of Oxford, during his visit in 2019–20

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  • We are concerned with a two-dimensional Riemann problem for the pressure gradient system that is a hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shocks and two vortex sheets are generated between the adjacent states. The solutions keep the four constant states and four planar waves outside the outer sonic circle in the self-similar coordinates, while the two shocks keep planar until meeting the outer sonic circle at two different points and then generate a diffracted shock to connect these points, whose location is apriori unknown. Then the problem can be formulated as a free boundary problem, in which the diffracted transonic shock is the one-phase free boundary to connect the two points, while the other part of the sonic circle forms a fixed boundary. We establish the global existence of a solution and the optimal Lipschitz regularity of both the diffracted shock across the two points and the solution across the outer sonic boundary. Then this Riemann problem is solved globally, whose solution contains two vortex sheets and one global shock containing the two originally separated shocks generated by the Riemann data.

    Mathematics Subject Classification: Primary: 35L65, 35M10, 35M12, 35R35, 35B36, 35L67; Secondary: 76L05, 76N10, 35D30, 35J67, 76G25.


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  • Figure 1.  The general Riemann initial data

    Figure 2.  The configuration of the four initial waves

    Figure 3.  The Riemann data and the global solution when $ \alpha_1 = 0 $

    Figure 4.  Hypothetical curves

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